Tag: maths

Questions Related to maths

The point $A(2, 1)$ is translated parallel to the line $x- y = 3$ by a distance $4$ units. If the new position $A'$ is in third quadrant, then the coordinates of $A'$ are

combining transformations transformations vectors and transformations maths

  1. $(2 + 2 \sqrt{2}, 1 + 2\sqrt{2})$

  2. $(-2 + \sqrt{2}, -1 -2 \sqrt{2})$

  3. $(2 - 2 \sqrt{2}, 1 - 2 \sqrt{2})$

  4. none of these

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Correct Option: C
Explanation:

Since the point $A(2, 1)$ is translated parallel to $x-y =3$, therefore $AA'$ has the same slope as that of $x-y =3$. Therefore, $AA'$ passes through $(2, 1)$ and has the slope of $1$. 

Here, $\tan  \theta = 1 $
$\Rightarrow \cos \theta = \displaystyle\frac{1 }{\sqrt{2}}, \sin \theta = \displaystyle \frac {1 }{ \sqrt{2}}$
Using the parametric form of the line, the coordinates of $A'$ can be written as $\left(2\pm \displaystyle\frac {4}{\sqrt 2}, 1\pm \displaystyle\frac{4}{\sqrt 2}\right) $. 
Now, $A'$ is in the third quadrant. 
Hence, the coordinates of $A'$ are $(2- 2 \sqrt{2}, 1 - 2 \sqrt{2})$.

If the axes are rotated through an angle of ${30}^{o}$ in the anti-clockwise direction, the coordinates of point $(4,-2\sqrt{3})$ with respect to new axes are-

combining transformations transformations vectors and transformations maths

  1. $(2,\sqrt{3})$

  2. $(\sqrt{3}, -5)$

  3. $(2,3)$

  4. $(\sqrt{3},2)$

Show answer
Correct Option: B
Explanation:

$x=r\cos \alpha=4$

$y=r\sin\alpha=-2\sqrt3$
$x^1=r\cos(\alpha-30^o)$
$y^1=r\sin(\alpha-30^o)$
$x^1=r[\cos\alpha\cos 30^o+\sin\alpha\sin 30^o]$
$x^1=\dfrac{4\sqrt{3}}{2}-2\sqrt{3}\times \dfrac{1}{2}$
$x^1=2\sqrt{3}-\sqrt{3}$
$\therefore x^1=\sqrt{3}$
$y^1=r[\sin\alpha\cos 30^o-\cos \alpha\sin 30^o]$
      $=-2\sqrt{3}\times \dfrac{\sqrt{3}}{2}-4\times \dfrac{1}{2}$
$y^1=-5$
$\therefore \alpha(\sqrt{3}, -5)$

Let $\displaystyle A=(1,0)$ and $\displaystyle B=(2,1).$ The line $AB$ turns about $A$ through an angle $ \dfrac{\pi}6$ in the clockwise sense, and the new position of $B$ is $B'$. Then $B'$ has the coordinates

combining transformations transformations vectors and transformations maths

  1. $\displaystyle \left ( \frac{3+\sqrt{3}}{2},\frac{\sqrt{3}-1}{2} \right )$

  2. $\displaystyle \left ( \frac{3\sqrt{3}}{2},\frac{\sqrt{3}+1}{2} \right )$

  3. $\displaystyle \left ( \frac{1-\sqrt{3}}{2},\frac{1+\sqrt{3}}{2} \right )$

  4. none of these

Show answer
Correct Option: A
Explanation:

Given, points are $A=(1,0)$ and $B=(2,1)$

Slope of $AB=\cfrac { 1-0 }{ 2-1 } =1$
Then angle of $AB$ with $x$-axis is 
$\angle BAX={ 45 }^{ 0 }$
Hence, $\angle B'AX={ 45 }^{ 0 }-{ 30 }^{ 0 }={ 15 }^{ 0 }$
Therefore for $B'\left( h,k \right) $
$h=1+\sqrt{2}\cos{ 15 }^{ 0 },k= \sqrt{2}\sin{ 15 }^{ 0 }\$
We have, $\sin \left(15^{0} \right) = \dfrac{\sqrt 6 -\sqrt 2 }{4} $
$\Rightarrow \cos \left (15^{0} \right) = \dfrac{\sqrt6 + \sqrt 2 }{4}$
$ \Rightarrow h=\cfrac { 3+\sqrt { 3 }  }{ 2 } ,k=\cfrac { \sqrt { 3 } -1 }{ 2 } $

The transformed equation of $3{ x }^{ 2 }+3{ y }^{ 2 }+2xy=2$. When the coordinate axes are rotated through an angle of $45$, is

combining transformations transformations vectors and transformations maths

  1. ${ x }^{ 2 }+2{ y }^{ 2 }=1$

  2. $2{ x }^{ 2 }+{ y }^{ 2 }=1$

  3. ${ x }^{ 2 }+{ y }^{ 2 }=1$

  4. ${ x }^{ 2 }+3{ y }^{ 2 }=1$

Show answer
Correct Option: B
Explanation:

Since, the axes are rotated through an angle $45$, then we replace $\left( x,y \right) $ by
$\left( x\cos { 45 } -y\sin { 45 } ,x\sin { 45 } +y\cos { 45 }  \right) $
Given equation is $3{ x }^{ 2 }+3{ y }^{ 2 }+2xy=2$
Therefore, $ 3{ \left( \dfrac { x }{ \sqrt { 2 }  } -\dfrac { y }{ \sqrt { 2 }  }  \right)  }^{ 2 }+3\dfrac { x+y }{ \sqrt { 2 }  } +2\dfrac { x-y }{ \sqrt { 2 }  } \dfrac { x+y }{ \sqrt { 2 }  } =2$
$\Rightarrow \dfrac { 3 }{ 2 } \left( { x }^{ 2 }+{ y }^{ 2 }+2xy \right) +\dfrac { 3 }{ 2 } \left( { x }^{ 2 }+{ y }^{ 2 }-2xy \right) +\dfrac { 2 }{ 2 } \left( { x }^{ 2 }-{ y }^{ 2 } \right) =2$
$\Rightarrow  4{ x }^{ 2 }=2{ y }^{ 2 }=2$
$\Rightarrow  2{ x }^{ 2 }+{ y }^{ 2 }=1$

$\overset \leftrightarrow{PQ}$ is perpendicular to $\overset \leftrightarrow{RS}$ is symbolically written as

maths drawing of different geometrical figures constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $\overset \leftrightarrow{PQ}\, \perp \, \overset \leftrightarrow{RS}$

  2. $\overset \leftrightarrow{PQ}\, \parallel \, \overset \leftrightarrow{RS}$

  3. $\overset \leftrightarrow{PQ}\, \neq \, \overset \leftrightarrow{RS}$

  4. $\overset \leftrightarrow{PQ}\, = \, \overset \leftrightarrow{RS}$

Show answer
Correct Option: A
Explanation:

$ \overleftrightarrow { PQ } $ is perpendicular to $ \overleftrightarrow { RS } $ is symbolically written as $ \overleftrightarrow { PQ } \bot  \overleftrightarrow { RS }  $

When two line segments meet at a point forming right angle they are said to be __________ to each other.

maths drawing of different geometrical figures constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. Parallel

  2. Perpendicular

  3. Equal

  4. None of the above

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Correct Option: B
Explanation:

When two line segments meet at a point forming right angle they are said to be perpendicular to each other.

$\displaystyle \overleftrightarrow {PQ}$ is perpendicular to $\displaystyle \overleftrightarrow {RS}$ is symbolically written as:

maths drawing of different geometrical figures constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $\displaystyle \overleftrightarrow {PQ}\perp \overleftrightarrow {RS}$

  2. $\displaystyle \overleftrightarrow {PQ}\parallel \overleftrightarrow{RS}$

  3. $\displaystyle \overleftrightarrow {PQ}\neq \overleftrightarrow{RS}$

  4. $\displaystyle \overleftrightarrow{PQ}= \overleftrightarrow {RS}$

Show answer
Correct Option: A
Explanation:

$ \overleftrightarrow { PQ } $ is perpendicular to $ \overleftrightarrow { RS } $ is symbolically written as $\overleftrightarrow { PQ } \bot  \overleftrightarrow { RS } $

When two lines are perpendicular to each other, the angle is said to be _______ angle.

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. acute

  2. right

  3. obtuse

  4. equal

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Correct Option: B
Explanation:

Two given lines are perpendicular means the angle between them is $90^o$, i.e. a right angle.

A perpendicular is drawn using

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line

  1. scale

  2. scale protractor

  3. set square

  4. divider

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Correct Option: B
Explanation:

A perpendicular is drawn using scale, protractor as well as set squares.

When a perpendicular is drawn to a given line, in what ratio is the line divided into?

maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

  1. $1:1$

  2. $1:2$

  3. $2:1$

  4. Cannot be said

Show answer
Correct Option: D
Explanation:

A line does not have a definite length.

Hence, when a perpendicular is drawn to the given line, nothing can be said about the ratio it gets divided into.